Adaptive discrete-time sliding mode controller

ABSTRACT

An adaptive, discrete-time sliding mode controller (SMC) is disclosed which detects and adapts to gain variations in the controlled plant. The overall control effort is generated by combining a linear control effort with a discrete-time sliding mode control effort generated by switching between gains in order to drive the system&#39;s phase states toward a sliding line trajectory. A sliding mode variable σ k  defines the position of the system phase states relative to the sliding line. The SMC controller is designed such that the sliding mode variable σ k  crosses the sliding line and changes sign at every sample interval. For the nominal plant gain, the SMC controller is also designed such that the magnitude of the sliding mode variable σ k+1 =−σ k  will remain constant (σ k+1 =−σ k ) and substantially constrained to 
     
       
         |σ k |=Δ/( 1 +λ) 
       
     
     where Δ and λ are predetermined design constants. The deviation in if the magnitude of σ k  from nominal is a direct measure of the deviation in the plant gain. Accordingly, the present invention measures the error between the measured magnitude of σ k  and the nominal magnitude, and uses this measured error to adapt to variations in the plant gain.

RELATED APPLICATIONS

This application is related to U.S. patent application Ser. No. 08/943,997 entitled “Improved Chatter Reduction in Sliding Mode Control of a Disk Drive Actuator” which is hereby incorporated by reference. Now U.S. Pat. No. 5,847895 as of Dec. 8, 1998.

FIELD OF INVENTION

The present invention relates to control systems technology, and more specifically, to an adaptive, discrete-time sliding mode controller for controlling the motion of mechanical apparatus such as a read head in a disk storage system.

BACKGROUND OF THE INVENTION

Sliding mode control (SMC) is a relatively new development in control systems which was first developed in Russia in the 1950's. It is a subclass of Variable Structure Control (VSC), control systems that switch between structures in a non-linear manner in order to drive phase states of the system toward a phase plane trajectory. The phase states of the system include the position error and its derivatives (velocity, acceleration, etc.); the location of the phase states within a phase plane define the state of the system at any given time, and the movement of the phase states through the phase plane is referred to as the phase state trajectory.

When the SMC is operating in a first structure, the phase states follow a first phase state trajectory, and when the SMC is operating in a second structure, the phase states follow a second phase state trajectory. By switching between the first and second structures, the phase states are driven toward a third phase state trajectory, referred to as the sliding line (or hperplane for higher order systems), defined within the phase plane where the first and second phase state trajectories intersect in opposite directions. The switching action is controlled by the location of the phase states relative to the sliding line; when the phase states cross the sliding line while following the first phase state trajectory, the system switches to the second structure to drive the phase states toward the sliding line by following the second phase state trajectory. In this manner, the phase states continuously switch across the sliding line as they follow the sliding line toward the origin of the phase plane (i.e., sliding mode). SMC has the advantage in that the closed loop response is defined by parameters in the controller and it is substantially insensitive to parameter variations in the plant and external load disturbances.

Initially, SMC systems were developed and implemented in continuous-time wherein the phase states are continuously monitored such that the system switches structures instantaneously when the phase states cross the sliding line. An inherent problem with this approach is that the continuous switching action may induce undesirable noise in the system (electrical and acoustic) and it may excite modelled (as well as unmodelled) system dynamics. The above referenced co-pending patent application entitled “Improved Chatter Reduction in Sliding Mode Control of a Disk Drive Actuator” discloses a method for reducing the amount of switching noise by defining a boundary layer around the sliding line and switching between structures only when the phase states exceed the boundary layer. Although this technique reduces the switching noise, there are other inherent problems with continuous-time SMC. Namely, to achieve the desired robustness to parameter variations and external disturbances, it can require gains in the individual structures that exceed the control effort limitations (e.g., exceed the available drive current).

Discrete-time SMC is a more recent development which addresses the drawbacks of continuous-time SMC by combining a conventional linear control effort with a discrete-time sliding mode control effort. This approach is discussed by Weibing Gao in “Discrete-Time Variable Structure Control Systems,” IEEE Transactions on Industrial Electronics, Vol. 42, No. 2, April 1995. With discrete-time SMC, the phase states are monitored in discrete-time and the system switches between structures at the sampling rate rather than continuously. Thus, there is an inherent boundary layer about the sliding line with a width defined by the sampling period, as well as other parameters of the controller. Rather than switch continuously such that the phase states “slide” along the sliding line, the phase states switch across the sliding line in a zigzag manner while sliding toward the origin of the phase plane. Another characteristic of discrete-time SMC is that nominally the phase states will cross the sliding line at every sampling period, and the absolute magnitude of the distance between the phase states and the sliding line will remain within the inherent boundary layer.

The linear control effort in discrete-time SMC decreases the necessary switching control effort while still maintaining a certain degree of insensitivity to parameter variations and external load disturbances. However, the system's insensitivity to parameter variations depends on the width of the boundary layer; the system's robustness can be increased by increasing the boundary layer, but this requires an increase in the switching control effort and eventually leads to the same problems associated with continuous-time SMC. Increasing the boundary layer also limits the maximum slope of the sliding line since the sliding line must be constrained within a particular region of the phase plane in order to maintain global stability.

There is, therefore, the need for an improved discrete-time sliding mode controller that exhibits the desired insensitivity to parameter variations and external load disturbances, without increasing the switching noise and without requiring a large control effort in the switching term. A further object of the present invention is to provide a discrete-time sliding mode controller wherein the slope of the sliding line can be increased to a more optimal value within the phase plane while maintaining global stability.

SUMMARY OF THE INVENTION

An adaptive, discrete-time sliding mode controller (SMC) is disclosed which detects and adapts to gain variations in the controlled plant. The overall control effort is generated by combining a linear control effort with a discrete-time sliding mode control effort generated by switching between gains in order to drive the system's phase states toward a sliding line trajectory. A sliding mode variable σ_(k) defines the position of the system phase states relative to the sliding line. The SMC controller is designed such that the sliding mode variable σ_(k) crosses the sliding line and changes sign at every sample interval. For the nominal plant gain, the SMC controller is also designed such that the magnitude of the sliding mode variable σ_(k) will remain constant (σ_(k+1)=−σ_(k)) and substantially constrained to

|σ_(k)|=Δ/(1+λ)

where Δ and λ are predetermined design constants. The deviation in the magnitude of σ_(k) from nominal is a direct measure of the deviation in the plant gain. Accordingly, the present invention measures the error between the measured magnitude of σ_(k) and the nominal magnitude, and uses this measured error to adapt to variations in the plant gain.

By adapting to the plant gain in a manner that maintains the magnitude of σ_(k) to the nominal value of Δ/(1+λ), the width of the boundary layer about the sliding line can be decreased while maintaining the desired insensitivity to parameter variations. Furthermore, because the boundary layer about the sliding line is minimized, the slope of the sliding line can be increased to a more optimal value while maintaining global stability.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of the disk drive control system of the present invention.

FIG. 2A and 2B show an example data format of a magnetic disk comprising a plurality of concentric data tracks with embedded servo fields.

FIG. 3A shows an example second order system controlled by a sliding mode controller designed using the continuous time existence equation.

FIG. 3B are the phase plane plots for the position error and error velocity phase states for the positive and negative feedback modes of the control system shown in FIG. 3A.

FIG. 4 illustrates the operation of the sliding mode controller in driving the phase states toward a predetermine linear phase trajectory.

FIG. 5A is a block diagram of the discrete-time sliding mode controller of the present invention comprising a linear control signal generator and a sliding line signal generator.

FIG. 5B shows details of the linear control signal generator of FIG. 5A, including a means for estimating the pseudo derivative of the reference signal.

FIG. 5C shows details of the sliding line signal generator of FIG. 5A.

FIG. 6 illustrates the zigzag switching action of the sliding mode variable σ_(k) and the effect on the magnitude of σ_(k) when the plant gain changes.

FIG. 7 illustrates various regions of operation which defines when the sliding mode variable σ_(k) will cross the sliding line on every sample period, and when the system will become unstable.

FIG. 8A is a block diagram of the discrete-time sliding mode controller comprising an adaptive processor for adaptively adjusting the gain of the plant by minimizing the deviation in the magnitude of σ_(k) from nominal.

FIG. 8B shows details of the adaptive processor of FIG. 8A.

FIG. 9 is a flow diagram executed by the adaptive processor of FIG. 8A.

FIG. 10 is a plot of the analytical versus simulated values of the magnitude of the sliding mode variable σ_(k) over a range of plant gain variations.

FIGS. 11A-11D illustrate the step response of the system for various values of plant gain.

FIG. 12 is a plot illustrating chaotic onset of the discrete-time SMC where the sliding mode variable σ_(k) does not cross the sliding line on successive sample periods if the plant gain is too low.

FIG. 13 is a plot of the chaotic boundary for various values of the slope of the sliding line c.

FIGS. 14A-14C show the response of the system to various step values when the system is operating near the chaotic boundary.

FIG. 15A shows a first embodiment for estimating the reference signal in a disk storage device by minimizing the position error phase state.

FIG. 15B shows an alternative embodiment for estimating the reference signal in a disk storage device by minimizing the deviation from nomnal in the magnitude of the sliding mode variable σ_(k).

FIG. 15C illustrates a method for estimating an unknown, non-repeatable reference by minimizing the deviation from nominal in the magnitude of the sliding mode variable σ_(k).

DETAILED DESCRIPTION OF THE DRAWINGS System Overview

In disk storage systems, such as magnetic and optical disk drives, a write transducer or write head writes digital data onto a disk storage medium in response to a write command received from a host system. The digital data serves to modulate a current in a write coil or an intensity of a laser beam so that a sequence of corresponding magnetic flux or optical transitions are written onto the disk medium in a series of concentric or spiral data tracks. To read this recorded data, a read transducer or read head passes over the disk medium and transduces the magnetic or optical transitions into pulses in an analog signal. These pulses are then decoded by read channel circuitry to reproduce the digital data.

The read head is typically actuated over the disk by means of a voice coil motor (VCM). In magnetic disk storage devices, the read head comprises a transducing coil suspended by a load beam and actuated radially over the disk by a rotary type VCM. In optical disk storage devices, the read head comprises optics and photodetectors housed in a sled assembly actuated by a two stage servoing mechanism: a first stage VCM actuator slides the sled assembly along a lead screw for coarse positioning the read head over a data track while seeking, and a second stage VCM actuator rotates an optical lens radially for fine positioning while tracking.

The servo system performs two functions: a “seek” or “access” function in which the servo system moves the head to a selected track; and when the head reaches the selected track, the servo system commences a “track following” or “tracking” function in which it accurately positions the head over a centerline of the track and maintains that position as successive portions of the track pass by the head. Servo control information embedded within the data provides inter-track head position information so that a head position error, indicative of a difference between the estimated head position and the desired head position, can be computed. In response to the head position error, the servo control system generates a VCM control signal to align the head over the centerline of the selected track, thereby driving the head position error to zero.

FIG. 1 is an overview of the disk drive control system of the present invention. A spin motor 2 spins a magnetic or optical disk 4 with computer data recorded thereon under a read head 6. An actuator 8 positions the read head 6 over a selected track, and a voice coil motor (VCM) 10 controls the motion of the actuator 8. Read channel circuitry 12, connected to receive the analog signal from the read head 6, demodulates the digital data recorded on the disk and transmits servo information to a state estimator 14. The state estimator 14 models the VCM control system in order to generate an estimated actuator position 16 that is subtracted from a reference actuator position 18 at adder 20 to produce an actuator position error X1 22. The reference actuator position 18 is the position of a selected track at which data is to be read (plus a runout signal 28 described below). A sliding mode controller 26, responsive to the actuator position error X1 22, computes an acceleration motor control signal U 28, amplified by amplifier 30, and applied to the VCM 10 to position the read head over the selected track. The state estimator 14 also processes the motor control signal U 28 to generate the estimated actuator position 16.

The state estimator 14 filters out errors in the servo position information caused by noise in the recording channel. State estimators are well known in the prior art such as described in U.S. Pat. No. 4,679,103, the disclosure of which is hereby expressly incorporated by reference. There are also other well known techniques other than state estimators for generating the actuator phase states for use by the present invention.

Referring now to FIGS. 2A and 2B, shown is an exemplary data format of a typical magnetic disk storage medium 4 comprising a plurality of concentric data tracks with an inner data track 32 and an outer data track 34. Each data track comprises a plurality of sectors 36 and a plurality of servo fields 38 embedded therein. The servo fields 38 comprise track number information 40 processed by the state estimator 14 to generate the estimated actuator position 16 while seeking the selected track. Once the read head slows to within a predetermined velocity as it approaches the selected track, the state estimator 14 processes the track number 40 and fine position information, such as servo bursts 42 in the servo fields 38, to align the head over the centerline of the selected track while reading the data. Servo bursts are commonly employed in magnetic disk storage devices and comprise a plurality of pulses recorded at precise intervals and offsets from the track centerline. The magnitude of these burst pulses sensed by the read head indicates the off track position of the head. In optical disk storage devices, the off track position information is derived by sensing the read signal's offset from the center of the photodetector.

In both magnetic and optical disk storage devices, the eccentricities of the disk storage medium may result in a sinusoidal reference signal referred to as runout. In FIG. 1, runout is modelled in the system by adding 24 a runout signal 25 to the position reference signal 27 representing the static centerline of the selected track. Thus, the runout is modelled as a component of the actual reference signal 18 used to generate the read head position error phase state X1 22. As described in greater detail below, it is necessary to derive an estimate of the derivative of the runout signal 25 (if not negligible) in order to enable operation of the sliding mode controller 26 of the present invention.

Theory of Operation

Operation of a conventional continuous-time sliding mode controller is understood with reference to FIG. 3A. Shown is an example second order system 44 controlled by an example sliding mode controller 45 that switches 46 between a positive gain 47 and negative gain 48 not necessarily equal in magnitude. A position error X1 49 is generated at the output of adder 50 by subtracting an estimated position 51 of the controlled system from a desired position reference command 52. The position error X1 49 is multiplied by the positive gain 47 or the negative gain 48 depending on the state of switch 46 to generate an acceleration command U 53 as the input to the controlled system 44. The output of integrator 54 is the velocity of the controlled system 44 which is also the negative of the position error velocity −X2 55. When switch 46 selects the positive gain 47, it is a negative feedback system, and when switch 46 selects the negative gain 48, it is a positive feedback system. In their individual structures the system is unstable; however, the system can be made stable by repeatedly switching between the two structures.

The state space equation in negative feedback is $\begin{matrix} {{\frac{{X1}}{t} = {X2}}{\frac{{X2}}{t} = {K\quad {X1}}}} & (1) \end{matrix}$

The general solution to equation (1) is

X1=A sin(Kt+φ)  (2)

X2={overscore (K)} A cos(Kt+φ)  (3)

Combining equations (2) and (3), $\begin{matrix} {{\frac{{X1}^{2}}{A^{2}} + \frac{{X2}^{2}}{{KA}^{2}}} = 1} & (4) \end{matrix}$

The phase plane plot of equation (4) is a set of ellipses with eccentricities A and {overscore (K)}A as illustrated in FIG. 3B.

The state space equation in positive feedback is $\begin{matrix} {{\frac{{X1}}{t} = {X2}}{\frac{{X2}}{{t}.} = {{+ K}\quad {X1}}}} & (5) \end{matrix}$

The solution to equation (5) is

X1=B1e^({overscore (K)}t)+B2e^(−{overscore (K)}t)

X2={overscore (K)}B1e^({overscore (K)}t)−K B2e^(−{overscore (K)}t)  (6)

Combining equations (5) and (6), $\begin{matrix} {{\frac{{X1}^{2}}{4{B1B2}} - \frac{{X\quad}^{2}}{4\quad {KB1B2}}} = 1} & (7) \end{matrix}$

The phase plane plot of equation (7) is a set of hyperbolas with two asymptotes as shown in FIG. 3B.

The two individual phase plane trajectories of equations (4) and (7) result in an unstable system since the phase states never reach the origin. It is possible, however, to reach the origin by driving the phase states along a third phase trajectory defined at the intersection of the negative and positive feedback trajectories. This is achieved by switching between the positive and negative gains in response to the current phase state values so that the phase states follow the predetermined third phase trajectory.

The switching operation is understood with reference to FIG. 4 where the predetermined third phase trajectory is shown as a linear segment 60. When a new track is selected, the initial actuator position error is at point A, and the control system is initially switched to select the positive gain (i.e., negative feedback). As the actuator begins to accelerate toward the selected track, the phase states follow the arc trajectory 64 of the negative feedback mode. When the phase states reach the beginning of the third phase trajectory 60 at the intersection point B, the sliding mode controller switches to the negative gain and the phase states begin to follow the hyperbola trajectory 66 of the positive feedback mode. When the phase states cross the third phase trajectory 60 at point C, the controller switches back to the positive gain to drive the phase states along arc 68 back toward the third phase trajectory 60. This switching action is repeated so that the phase states slide along the linear segment 60 toward the origin of the phase plane. When the phase states are within a predetermined minimum distance from the origin of the phase plane, the system switches to a tracking mode where the sliding mode controller 26 repeatedly switches between positive and negative feedback in order to maintain the phase states near the origin of the phase plane, thereby aligning the read head 6 over the centerline of the selected track while reading the data.

The time domain response of the control system when the phase states follow the sliding line of FIG. 4 is

X1(t)=X1(t1)e^(−C(t−t1))  (8)

where t1 is the time when the phase states reach the sliding line at point B. Equation (8) is the average response of the system along the sliding line and it is substantially unaffected by variations in the parameter K or by external load disturbances. Consequently, it is not necessary to know the exact parameters of the system in order to determine the value for K. Equation (8), together with the existence Equation (11) below, demonstrate that the system's response is first order and globally stable within the sliding mode region.

The sliding mode controller determines when to switch between the positive and negative gains by observing the phase state values. The linear phase trajectory of FIG. 4 is defined by

X2=−CX1  (9)

where the constant C is the slope of the linear segment 60. By observing the phase states, the sliding mode controller switches the gains so that

σ=X2+CX1=0.  (10)

The sliding mode controller switches to the positive gain when σX1>0 and to the negative gain when σX1<0 in order to drive the phase states toward the sliding line.

The overall response of the system is made faster by increasing the slope of the sliding line (i.e., increasing C). However, an important limitation in sliding mode control is that the sliding must be constrained to a region in the phase plane where the positive and negative feedback phase trajectories intersect in opposite directions. From FIG. 4 it follows that the slope of the sliding line must be constrained to 0<C<{overscore (K)}. A further relationship derived from this constraint is $\begin{matrix} {{\lim\limits_{\sigma\rightarrow 0}{\sigma \frac{\sigma}{t}}} < 0.} & (11) \end{matrix}$

Equation (11) is known as the existence equation and is used to determine values for the positive and negative gains.

Discrete-Time Sliding Mode Control

The present invention overcomes the drawbacks of sliding mode control which operates according the above-described continuous-time existence equation (11) by employing discrete-time sliding mode control. Consider a discrete-time system

X_(k+1)=ΦX_(k)+ΓU_(k)  (17)

where X is a state matrix representing the phase states of the system (position, velocity, acceleration, etc.), Φ is a linear control matrix, and Γ is a scalar. A discrete-time control law is given by

σ_(k+1)=λσ_(k)−Δsgn(σ_(k))  (18)

where σ_(k) is a sliding mode variable computed from discrete-time phase state errors ε_(k), λ and Δ are predetermined constants, and sgn(x) returns the sign of x. The discrete-time phase state errors ε_(k), such as the position error, velocity error, etc., are computed by subtracting the estimated phase states X_(k) from a reference signal R_(k)

ε_(k)=R_(k)−X_(k)  (19)

or by a suitable means for estimating the phase state errors. The sliding mode variable σ_(k) of equation (18) is computed by multiplying the discrete-time phase state errors ε_(k) by a vector c which defines the sliding line in the phase plane

 σ_(k) =cε _(k.)  (20)

Substituting equation (20) into equation (18) yields

cε _(k+1) =λcε _(k)−Δsgn(σ_(k)).  (21)

Substituting equations (17) and (19) into equation (21) yields

c(R_(k+1)−(ΦX_(k)+ΓU_(k)))=λc(R_(k)−X_(k))−Δsgn(σ_(k)).

Solving for σ_(k) yields the control law for the servo problem

U_(k) =c(Φ−λI)(cΓ)⁻¹ε_(k) +c(R_(k+1)−ΦR_(k))(cΓ)⁻¹+Δ(cΓ)⁻¹sgn(σ_(k)).  (22)

Defining

Ψ=c(Φ−λI)(cΓ)⁻¹; and

Λ_(k+1)=C(R_(k+1)−ΦR_(k)) (cΓ)⁻¹;

yields

U_(k)=Φε_(k)+Λ_(k+1)+Δ(cΓ)⁻¹sgn(σ_(k)).  (23)

The control law (23) for the servo problem is a linear combination of the error states, a feed-forward term that requires knowledge of the reference, and a switching term based on the sliding line. Note that for the general regulator problem, the reference term Λ_(k+1) is normally set to zero for all states which is an accurate model for many systems such as magnetic disk storage systems.

Estimation of Λ_(k+1)

The term Λk+1 in the control law (23) approximates the derivative of the reference signal R_(k) and is thus called the pseudo derivative of the reference R_(k). For a large class of problems, Λ_(k+1) does not present a significant problem because the phase state trajectories of the system are well defined in an a priori sense, and the system trajectories are controlled in real time by manipulating Λ_(k+1). Robotic manipulators are a good example of such systems. However, for a wide class of servo systems direct access to the reference is not available, and only information at the error node(s) may be used to estimate the apparent reference. Magnetic and optical disk storage devices are examples of such systems.

Equation (22) can be rewritten as

Λ_(k+1)=U_(k) −cΦ(cΓ)⁻¹ε_(k)+λ(cΓ)⁻¹σ_(k)−Δ(cΓ)⁻¹sgn(σ_(k)).  (24)

Using the discrete-time existence equation (18), equation (24) can be written as

Λ_(k+1)=U_(k) −cΦ(cΓ)⁻¹ε_(k)+(cΓ)⁻¹σ_(k+1.)

For most systems where the reference is changing slowly compared to the sample rate, Λ_(k+1) may be estimated as

Λ_(k+1)≈Λ_(k)=U_(k−1) −cΦ(cΓ)⁻¹ε_(k−1)+(cΓ)⁻¹σ_(k).  (25)

The accuracy of equation (25) can be improved using a Taylor series expansion

Λ_(k+1)≈Λ_(k)+(Λ_(k)−Λ_(k−1))/T.  (26)

Equation (25) is a function of the prior control effort, the prior error, and the current value of σ. Other methods could be employed to estimate Λ_(k+1), including Kalman Estimators.

Hardware Description

FIG. 5A is a block diagram of the sliding mode controller 26 of the present invention for use in a second order system (the aspects of the present invention can easily be extended to higher order systems). In this embodiment, the sliding mode controller 22 is responsive to two phase state errors, ε1(k) and ε2(k), which may represent a position error and a velocity error of the actuator 8. Various methods are known in the art for generating phase state errors, including the state estimator 14 shown in FIG. 1. The phase state errors are input into a linear controller 90 and a sliding line controller 92, and their outputs are combined at adder 94 to implement the control effort U_(k) of equation (23). The sliding line controller 92 generates the sliding line variable σ_(k) which is input into the linear controller 90 for use in computing the pseudo derivative reference term Λ_(k+1).

Details of the linear controller 90 of FIG. 5A are shown in FIG. 5B. The error phase states are multiplied 96 by

c(Φ−λI) (cΓ)⁻¹

to generate the term

Ψε_(k)

of equation (23). To generate an estimate of the pseudo derivative reference term Λ_(k+1), the phase state errors are delayed through delay elements 98A and 98B and then multiplied 100 by

cΦ(cΓ)⁻¹

to generate the term

cΦ(cΓ)⁻ε_(k−1)

of equation (25). The sliding line variable σ_(k) is multiplied 102 by

 (cΓ)⁻¹

to generate the term

(cΓ)⁻¹σ_(k)

of equation (25). The control effort U_(k) is delayed by delay element 104 and the estimated pseudo derivative reference (PDR) term Λ_(k) generated at the output of adder 106 thereby implementing equation (25). The first difference 108 of Λ_(k) is computed and added 110 to Λ_(k) to generate the estimated PDR term Λ_(k+1) according to equation (26). The estimated PDR term Λ_(k+1) is then added 112 to the output of multiplier 96 to generate the term

Ψε`k+Λ_(k+1)

of equation (23).

Details of the sliding line controller 92 of FIG. 5A are shown in FIG. 5C. The phase state errors are multiplied 114 by the sliding line c where c defines the phase state trajectory in the phase plane as described above. For example, in equation (16) the sliding line c is defined as

c=[1, C3].

The multiplication of the phase state errors by the sliding line c generates the sliding line variable σ_(k) per equation (20). The sign 116 of the sliding line variable σ_(k) is multiplied 118 by

μ=Δ(cΓ)⁻¹

to generate the term

μsgn(σ_(k))

of equation (23).

σ_(k) Steady State

In the control law of equation (18), σ_(k) is non-increasing and changes sign on successive sample periods. When σ_(k)<0, to maintain stability and simultaneous switching it requires

σ_(k+1)≦−σ_(k).

From the above equations

cε_(k+1) ≦−cε_(k);

c(R_(k+1)−(ΦX_(k)+ΓU_(k)))≦−cε_(k.)

Solving for U_(k) yields

U_(k) ≧c(Φ+I)(cΓ)⁻¹ε_(k) +c(R_(k+1)−ΦR_(k))(cΓ)⁻¹.  (27)

However, an expression for U_(k) has already been derived in equation (22). Substituting equation (22) into (27) yields

c(Φ−λI)(cΓ)⁻¹ε_(k) +c(R_(k+1)−ΦR_(k))(cΓ)⁻¹+Δ(cΓ)⁻¹sgn(σ_(k))≧c(Φ+I)(cΓ)⁻¹ε_(k) +c(R_(k+1)−(ΦR_(k))(cΓ)⁻¹

Combining terms and simplifying leads to

(−λcε _(k)+Δsgn(σ_(k))≧cε _(k)

which leads to

Δ/(1+λ)sgn(σ_(k))≧σ_(k)

A similar result follows for σ_(k)>0 with the inequality reversed, which leads to

−Δ/(1+λ)≦σ_(k)≦Δ/(1+λ).  (28)

Equation (28) specifies that the absolute magnitude of the sliding mode variable σ_(k) will be constrained within a boundary layer such that

 |σ_(k)|≦Δ/(1+λ)

The width of the boundary layer is a function of the Δ, λ and the sampling rate. The width of the boundary layer is generally defined through selecting an appropriate value for Δ, whereas the time constant of the loop is generally defined through appropriate selection of λ and the slope of the sliding line c. The system's insensitivity to parameter variations and load disturbances is increased by increasing the width of the boundary layer (i.e., increasing Δ), the trade-off being an increase in the required control effort and an increase in switching noise.

As described in greater detail below, the servo system can be adapted in real time so as to maintain the optimal condition where the magnitude of the sliding mode variable σ_(k) in steady state remains constant (i.e., σ_(k+1)=−σ_(k)) while satisfying the discrete-time switching condition of crossing the sliding line at every sample period. This maintains a constant, and arbitrarily small boundary layer that in turn minimizes the switching noise.

Design of the Sliding Line

The steady state behavior of σ_(k) as described above can be used to determine the optimal value for the sliding line c. Multiplying both sides of equation (28) by ε_(k) ^(T) yields

σ_(k)ε_(k) ^(T) =cε _(k)ε_(k) ^(T)=Δ/(1+λ)sgn(σ_(k))ε_(k) ^(T).

Taking the expectation E of both sides with E(ε_(k) ^(T))=0, yields the matrix equation

cE(ε_(k)ε_(k) ^(T))=0.  (29)

Expanding equation (29) and assuming a second order system without loss of generality, yields the simultaneous equations

c ₁E(ε_(k) ¹ε_(k) ¹)+E(ε_(k) ¹ε_(k) ²)=0

c ₁E(ε_(k) ²ε_(k) ¹)+E(ε_(k) ²ε_(k) ²)=0

For a stationary or deterministic process

E(ε_(k) ¹ε_(k) ²)=E(ε_(k) ²ε_(k) ¹).

Solving the simultaneous equations results in an expression for c

c₁ ²=E(ε_(k) ²ε_(k) ²)/E(ε_(k) ¹ε_(k) ¹).  (30)

Equation (30) illustrates that the slope of the sliding line c is a function of the ratio between the rms error of the two states (i.e., the ratio of the rms position error over the rms velocity error) for a second order system. The expected rms position error is generally one of the design criteria of a system, and is therefore known (it is usually specified as a ratio between the reference and the error at specific reference frequencies). The rms velocity error will follow as a result of the system model.

Adapting to Plant Gain Variations

The present invention uses the discrete-time existence equation (18) together with the sliding mode condition that σ_(k) cross the sliding line at every sample instance in order to adapt the controller to compensate for gain variations in the plant due, for example, to temperature or parameter drift. Adapting to gain variations in the plant increases the system's insensitivity to parameter variations without having to increase the boundary layer and attendant switching noise. Furthermore, because the boundary layer is minimized, the slope of the sliding line can be increased to a more optimal value while maintaining global stability.

Expressing the plant gain variation as β (with β=1 nominal), the discrete-time system (17) can be rewritten as

X_(k+1)=ΦX_(k)+βΓU_(k).

From equations (22) and (23), algebraic manipulations leads to the discrete-time error equation

ε_(k+1)=ε_(k)(Φ−βΓΨ)+(I−βΓc(cΓ)⁻¹)(R_(k+1)−ΦR_(k))−βΓΔ(cΓ)⁻¹sgn(σ_(k)).  (31)

Multiplying by c and simplifying results in an expression for σ

σ_(k+1) =c(1−β)(R_(k+1)−ΦR_(k))+(1−β) cΦε_(k)+β(λσ_(k)−Δsgn(σ_(k))).  (32)

The third term in equation (32) is equivalent to the nominal value of σ_(k+1) multiplied by the gain variation β and thus represents a direct measurement of the plant gain departure from nominal. The first and second terms of equation (32) represent movement away from the nominal gain of unity. This is understood with reference to FIG. 6 which shows the sliding line c and the nominal amplitude Δ/(1+λ) of the sliding mode variable σ_(k) according to equation (28). Thus, when the gain is nominal (β=1) the sliding mode variable σ_(k) will switch between positive and negative values at every sample period and remain on the dashed lines representing the sliding line boundary layer. If the plant gain decreases then the magnitude of the sliding mode variable σ_(k) will decrease, and if the plant gain increases the magnitude of the sliding mode variable σ_(k) will increase as shown in FIG. 6. Thus, one adaptive strategy is to adapt the loop gain in order to adjust the measured value of σ_(k) toward the nominal steady state value Δ/(1+λ) (i.e. toward the dashed lines in FIG. 6).

As discussed in greater detail below, if the gain of the plant drops below a certain threshold, then the sliding mode variable σ_(k) will not cross the sliding line and change sign on every sample period. Conversely, if the gain exceeds a certain threshold, then the system will become unstable. Thus, there are three regions of operation as illustrated in FIG. 7 which is a plot of the gain variance β versus the magnitude of the sliding mode variable σ_(k). In region I, the gain of the plant is below the lower threshold and σ_(k) does not switch across the sliding line at every sample period. In region III, the gain of the plant exceeds the upper threshold and the system is unstable. In the region between I and III (region II), the system is stable and the sliding mode variable σ_(k) crosses the sliding line and changes sign on every sample period. Thus, the system will operate normally and adapt to changes in the plant gain as long as the gain changes slow enough such that magnitude of σ_(k) remains within region II.

FIG. 8A is a block diagram of the adaptive, discrete-time sliding mode controller of the present invention. The sliding line variable σ_(k) output by the sliding line controller 92 is processed by an adaptive processor 120 which in turn adjusts the plant gain Kpa 30 according to the flow diagram of FIG. 9. First at step 122 a check is made to determine if the sliding line variable σ_(k) is crossing the sliding line and changing signs at every sample period. If not, then a loop is executed where the plant gain is incremented at step 124 until σ_(k) begins to switch at every sample interval. Then at step 126 a branch is executed based on the absolute magnitude of σ_(k). If the magnitude of σ_(k) is greater than nominal (i.e, . greater than Δ/1+λ), then at step 128 the plant gain is decreased incrementally. If at step 126 the magnitude of σ_(k) is less than nominal, then the plant gain is increased incrementally at step 130. Steps 128 and 130 are repeated until the magnitude of σ_(k) reaches the nominal value of Δ/1+λ.

Details of the adaptive processor 120 of FIG. 8A are shown in FIG. 8B. The adaptive processor 120 comprises: a switching detector 121 for determining whether σ_(k) is switching across the sliding line at every sample interval, an amplitude detector 123 for measuring the absolute magnitude of σ_(k), an error detector 125 for detecting an error between the measured amplitude of σ_(k) and the nominal amplitude, and a stability detector 127 for detecting when the system has the potential for going unstable (i.e., when σ_(k) crosses into region III in FIG. 7).

Many filtering options exist for implementing the switching detector 121; in FIG. 8B, a simple one-delay correlator is employed which operates according to the following equation $D_{k} = {\frac{1}{m}{\sum\limits_{j = 0}^{m}{{{sgn}\left( \sigma_{k - j} \right)}{{sgn}\left( \sigma_{k - j - 1} \right)}}}}$

where m is a predetermined integer. In the above equation, if all m prior values of σ_(k) have changed sign, then D_(k) would equal −1. If not, then D_(k)>−1. A threshold 129 may be employed in the switching detector 121 to compensate for noise; for example, −1<D_(k)>t where t is the threshold based upon anticipated noise. More filtering may be added if necessary by prior filtering σ_(k) prior to the correlator.

When σ_(k) is in region II or III of FIG. 7, it is necessary to 11 determine the magnitude for σ_(k) which is then compared to the nominal value to generate the error signal, as well as to determine stability of the system. In these regions, the magnitude of σ_(k) is either constant or increasing, therefore the first difference of σ_(k) is a convenient detector (other techniques, such as the rms value of sigma are also possible). If the first difference is constant, then operation in region II is assured. If the first difference is increasing, then σ_(k) is in region III. The amplitude detector 123 of FIG. 8B operates according to

Sam_(k)=|σ_(k)−σ_(k−1)|_(F)

where the subscript F denotes a filtering operation 131 on the absolute value 133 of the first difference 135. The filter is chosen based on the expected noise; a simple moving average filter with binary coefficients will generally suffice.

The error detector 125 of FIG. 8B generates the error between the measured amplitude of σ_(k) and the nominal amplitude; the error is then used in the adaptation algorithm as described above. The nominal value for the amplitude of σ_(k) is

Δ/(1+λ)

the error is detected as

err_(k)=2Δ/(1+λ)−|σ_(k)−σ_(k−1)|_(F).

The stability detector 127 of FIG. 8B determines when σ_(k) has crossed over into region III of FIG. 7 where the system may become unstable. Detecting when σ_(k) crosses into region III allows higher gains to be used in the adaptation algorithm. Furthermore, the adaptive, discrete-time sliding mode controller of the present invention can stabilize an unstable system when σ_(k) crosses into region III. The stability detector 127 is implemented by passing the measured amplitude Of σ_(k) through a simple 1-D low-pass filter, and comparing the output to a threshold 137. When the average amplitude of σ_(k) exceeds the threshold 137, it has crossed into region III of FIG. 7.

The output of the switching detector 121 and stability detector 127 are used to determine which of the three regions in FIG. 7 σ_(k) is currently in. Based on this decision, the gain of the adaptation loop is adjusted by selecting one of three predetermined gains through a multiplexer 139 as shown. The decision logic 141 and gain selected is illustrated in the following table

TABLE 1 Stability σ_(k) Sign Selected Detector Correlator Gain Unstable — G1 Stable Switching Every G2 Sample Stable Not Switching Every G3 Sample

The selected gain is then multiplied 143 by the error in σ_(k) and the result added 145 to the previous gain adjustment value β_(k−1) to generate the current gain adjustment value β_(k). The gain adjustment value β_(k) is then used to adjust the gain of the system amplifier 30 shown in FIG. 1.

The above-described adaptation strategy will work for small deviations of σ_(k) from nominal, but may not be adequate for cases where the deviation is significant since the first and second terms may dominate the third term in equation (32). To determine the regions where the adaptation strategy will work, it is first necessary to determine the steady state value of the errors ε_(k) and by inference σ_(k).

ε_(k) Steady State

The steady state errors ε_(k) can be determined using equation (31). Defining

Φ₀=Φ−βΓΨ

 A=I−βΓc(cΓ)⁻¹

B=βΓΔ(cΓ)⁻¹

equation (31) can be written as $\begin{matrix} \begin{matrix} {ɛ_{k + 1} = \quad {{\Phi_{0}ɛ_{k}} + {A\left( {R_{k + 1} - {\Phi \quad R_{k}}} \right)} - {{Bsgn}\left( \sigma_{k} \right)}}} \\ {ɛ_{k + 2} = \quad {{\Phi_{0}ɛ_{k + 1}} + {A\left( {R_{k + 2} - {\Phi \quad R_{k + 1}}} \right)} - {{Bsgn}\left( \sigma_{k + 1} \right)}}} \\ {= \quad {{\Phi_{0}^{2}ɛ_{k}} + {\Phi_{0}{A\left( {R_{k + 1} - {\Phi \quad R_{k}}} \right)}} - {\Phi_{0}{{Bsgn}\left( \sigma_{k} \right)}} +}} \\ {\quad {{A\left( {R_{k + 2} - {\Phi \quad R_{k + 1}}} \right)} - {{{Bsgn}\left( \sigma_{k + 1} \right)}.}}} \\ {ɛ_{k + 3} = \quad {{\Phi_{0}ɛ_{k + 2}} + {A\left( {R_{k + 3} - {\Phi \quad R_{k + 2}}} \right)} - {{Bsgn}\left( \sigma_{k + 2} \right)}}} \\ {= \quad {{\Phi_{0}^{3}ɛ_{k}} + {\Phi_{0}^{2}{A\left( {R_{k + 1} - {\Phi \quad R_{k}}} \right)}} - {\Phi_{0}^{2}{{Bsgn}\left( \sigma_{k} \right)}} +}} \\ {\quad {{\Phi_{0}{A\left( {R_{k + 2} - {\Phi \quad R_{k + 1}}} \right)}} - {\Phi_{0}{{Bsgn}\left( \sigma_{k + 1} \right)}} +}} \\ {\quad {{A\left( {R_{k + 3} - {\Phi \quad R_{k + 2}}} \right)} - {{{Bsgn}\left( \sigma_{k + 2} \right)}.}}} \\ {ɛ_{k + n} = \quad {{\Phi_{0}^{n}ɛ_{k}} + {\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}\left( {A\left( {\left( {R_{k + n - j} - {\Phi \quad R_{k + n - j - 1}}} \right) -} \right.} \right.}}}} \\ \left. \left. \quad {{Bsgn}\left( \sigma_{k + n - j - 1} \right)} \right) \right) \end{matrix} & (33) \end{matrix}$

Equation (33) may be simplified if we first consider the regulator problem where R=0. The steady state regulator error may be determined by taking the limit of the resulting equation $\begin{matrix} {{\lim\limits_{n\rightarrow\infty}ɛ_{k + n}} = {\lim\limits_{n\rightarrow\infty}\left\{ {{\Phi_{0}^{n}ɛ_{k}} - {\sum\limits_{j = 0}^{n - 1}{\left( {\Phi_{0}^{j}B} \right){{sgn}\left( \sigma_{k + n - j - 1} \right)}}}} \right\}}} & (34) \end{matrix}$

With the assumption that the sign of σ changes on successive sample intervals, equation (34) represents a geometric series which can be further reduced $\begin{matrix} {{\lim\limits_{n\rightarrow\infty}\left\{ {{\Phi_{0}^{n}ɛ_{k}} - {\sum\limits_{j = 0}^{n - 1}{\left( {\Phi_{0}^{j}B} \right){{sgn}\left( \sigma_{k + n - j - 1} \right)}}}} \right\}} = \quad {{\lim\limits_{n\rightarrow\infty}\left\{ {{\Phi_{0}^{n}ɛ_{k}} - {\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{B\left( {- 1} \right)}^{n - j - 1}}}} \right\}} = \quad {\lim\limits_{n\rightarrow\infty}\left\{ {{\Phi_{0}^{n}ɛ_{k}} - {\sum\limits_{j = 0}^{n - 1}{\left( {I - {\left( {- 1} \right)^{n - j}\Phi_{0}^{j + 1}}} \right)\left( {I - {\left( {- 1} \right)^{1}\Phi_{0}}} \right)^{- 1}B}}} \right\}}}} & (35) \end{matrix}$

With all elements of Φ₀ less than one, equation (35) reduces to $\begin{matrix} {{\lim\limits_{n\rightarrow\infty}ɛ_{k + n}} = {{{\lim\limits_{n\rightarrow\infty}\left\{ {\Phi_{0}^{n}ɛ_{k}} \right\}} + {\left( {I + \Phi_{0}} \right)^{- 1}B}} = {0 + {\left( {\left( {I + \Phi_{0}} \right)^{- 1}\beta} \right){{\Gamma\Delta}\left( {c\quad \Gamma} \right)}^{- 1}}}}} & (36) \end{matrix}$

The steady state value of σ may also be determined from equation (36). Multiplying by c and sgn(σ_(k)), σ_(k+1) is

σ_(k+1) =c(I+Φ₀)⁻¹βΓΔ(cΓ)⁻¹sgn(σ_(k)).  (37)

FIG. 10 shows the analytical solutions overlaid with the simulated values for σ steady state. Note that for certain threshold values of β less than one, the analytical solutions do not agree with the simulated values. These regions correspond to areas where the assumption of sgn(σ_(k+1))=−sgn(σ_(k)) is violated and equation (37) is no longer valid. FIGS. 11A-11D show a series of step responses with varying plant gain (beta).

Sufficient Conditions for Zero Crossing

The assumption that a will cross the sliding line and change sign at every sample period depends on the values selected for the sliding mode variables c, λ and Δ, as well as the plant transfer function and gain variation β. The values which satisfy this condition can be determined from equations (31) and equation (34). From equation (34), the error for the regulator problem (R=0) at state k+n is $ɛ_{k + n} = {{\Phi_{0}^{n}ɛ_{k}} - {\sum\limits_{j = 0}^{n - 1}{\left( {\Phi_{0}^{j}B} \right){{sgn}\left( \sigma_{k + n - j - 1} \right)}}}}$

From equation (31), the error at state k+n+1 is

ε_(k+n+1)=Φ₀ε_(k+n)−Bsgn(σ_(k+n)).

Therefore, for σ_(k+1)>0 a sign change will occur at every sample period if

cε _(k+n+1)<0

cΦ ₀ε_(k+n) −cBsgn(σ_(k+n))<0

cΦ ₀εk+n<cB.

A similar result can be derived for σ_(k+1)<0 resulting in the combined requirements of

−cB<cΦ ₀ε_(k+n) <cB.  (38)

Equation (38) is valid for any location of σ in the phase plane. Assuming equation (38) is satisfied such that σ changes sign at every sample period, equation (38) can be written as

−cB<cΦ ₀(I+Φ₀)⁻¹ <cB.  (39)

Expanding equation (38) using equation (34) and multiplying by (cB)⁻¹, equation (39) can be written as $\begin{matrix} {{- 1} < {\lim\limits_{n\rightarrow\infty}{\left\{ {{c\quad \Phi_{0}^{n + 1}ɛ_{k}} - {c{\sum\limits_{j = 1}^{n - 1}{\Phi_{0}^{j}{{Bsgn}\left( \sigma_{k + n - j - 1} \right)}}}}} \right\} ({cB})^{- 1}}} < 1} & (40) \end{matrix}$

Assuming that the initial condition term tends to zero, equation (40) can be written as

−1<_(n→∞)limc{Φ ₀ ¹B(sgn(σ_(k+n))+Φ₀ ²B(sgn(σ_(k+n+) ₁)+. . . ))}(cB)⁻¹<1  (41)

If successive values are non-increasing, then the series in equation (41) will converge to a bounded value between zero and one. The conditions for assuring that σ will cross the sliding line and change sign on every sample period is thus

cΦ ₀ ^(j+1)B<cΦ ₀ ^(j)B  (42)

FIG. 12 demonstrates equation (42) for a given system with c=[0.81] over a range of values for the plant gain (beta). When the plant gain β falls below a certain threshold, equation (42) is violated and the system becomes chaotic. The region where equation (42) is violated is primarily a function of the sliding line c and the gain β. FIG. 12 illustrates that the analytical estimates of steady state behavior are lower bounds to the potential steady state behavior. FIG. 13 shows the value of β that delineates the chaotic boundary for each value of c.

When equation (42) is violated and the system becomes chaotic, the steady state behavior will depend on the initial conditions of the system. It is possible that the steady state behavior, while stable, may be significantly different depending on initial conditions. FIGS. 14A-14C illustrate this by showing step responses of varying amplitudes while the system is in the chaotic region. Notice that in FIG. 14A and FIG. 14B when the amplitude of the step response is −0.1 and −0.2, a does not change sign on every sample period, whereas in FIG. 14C σ does change sign on every sample period when the amplitude of the step response is −0.5.

Extension to the Servo Problem

The above analysis of discrete-time sliding mode control applied to a regulator can be extended to a servo system having a non-zero reference signal. For σ_(k+1)>0, the above analysis with a non-zero reference signal leads to

cε _(k+n+1)<0

cΦ ₀ε_(k+n) +cA(R_(k+n+1)−ΦR_(k+n))−cBsgn(σ_(k+n))<0

cΦ ₀ε_(k+n) +cA(R_(k+n+1)−ΦR_(k+n))<cB.

A similar result can be derived for σ_(k+1)<0 resulting in the combined requirements of

−cB<cΦ ₀ε_(k+n) +cA(R_(k+n+1)−ΦR_(k+n))<cB.  (43)

Equation (43) is valid for all locations of σ in the phase plane, and it does not require that steady state behavior be attained. For the specific case of steady state behavior where σ crosses the sliding line and changes sign on every sample period, further simplification is possible $\quad {{\lim\limits_{n\rightarrow\infty}\left\{ {{c\quad \Phi_{0}ɛ_{k + n}} + {{cA}\left( {R_{k + n + 1} - {\Phi \quad R_{k + n}}} \right)}} \right\}} = \quad {{\lim\limits_{n\rightarrow\infty}\left\{ {{c\quad \Phi_{0}^{n + 1}ɛ_{k}} - {c\quad \Phi_{0}{\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{A\left( {R_{k + n - j} - {\Phi \quad R_{k + n - j - 1}}} \right)}}}}} \right\}} + \quad {({cA})\left( {R_{k + n + 1} - {\Phi \quad R_{k + n}}} \right)} - {\lim\limits_{n\rightarrow\infty}\left\{ {c\quad \Phi_{0}{\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{{Bsgn}\left( \sigma_{k + n - j - 1} \right)}}}} \right\}}}}$

Combining terms and taking the limit of the initial condition term leads to $\begin{matrix} {\quad {{\lim\limits_{n\rightarrow\infty}\left\{ {{c\quad \Phi_{0}ɛ_{k + n}} + {{cA}\left( {R_{k + n + 1} - {\Phi \quad R_{k + n}}} \right)}} \right\}} = \quad \quad {{\lim\limits_{n\rightarrow\infty}\left\{ {c\quad {\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{A\left( {R_{k + n + 1 - j} - {\Phi \quad R_{k + n - j}}} \right)}}}} \right\}} - {\lim\limits_{n\rightarrow\infty}\left\{ {c\quad \Phi_{0}{\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{{Bsgn}\left( \sigma_{k + n - j - 1} \right)}}}} \right\}}}}} & (44) \end{matrix}$

From equations (35) and (36), the second term in equation (44) reduces to

cΦ ₀(I+Φ₀)⁻¹Bsgn(σ_(k)).

For the servo problem, the residual derivative term modulates the switching action of σ. Equation (43) then simplifies to $\begin{matrix} {{- \left( {{cB} + {c\quad {\Phi_{0}\left( {I + \Phi_{0}} \right)}^{- 1}B}} \right)} < {\lim\limits_{n\rightarrow\infty}\left\{ {c{\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{A\left( {R_{k + n + 1 - j} - {\Phi \quad R_{k + n - j}}} \right)}}}} \right\}} < \left( {{cB} + {c\quad {\Phi_{0}\left( {I + \Phi_{0}} \right)}^{- 1}B}} \right)} & (45) \end{matrix}$

The magnitude of the middle term in equation (45) is more easily evaluated in the z domain $\begin{matrix} {{{{\lim\limits_{n\rightarrow\infty}\left\{ {c{\sum\limits_{j = 0}^{n - 1}{\Phi_{0}^{j}{A\left( {R_{k + n + 1 - j} - {\Phi \quad R_{k + n - j}}} \right)}}}} \right\}}} = {\left| {c\left( {{\Phi_{0}^{0}{A\left( {{zI} - \Phi} \right)}{R(z)}} + \quad {\Phi_{0}^{1}z^{- 1}{A\left( {{zI} - \Phi} \right)}{R(z)}} + {\Phi_{0}^{2}z^{- 2}{A\left( {{zI} - \Phi} \right)}{R(z)}} + \ldots} \right)} \right| = {{{{c\left( {\left( \frac{\Phi_{0}}{z} \right)^{0} + \left( \frac{\Phi_{0}}{z} \right)^{1} + \left( \frac{\Phi_{0}}{z} \right)^{2} + \ldots} \right)}{A\quad\left( {{zI} - \Phi} \right)}\quad {R(z)}}} = \quad \quad \left| \quad {{{cz}\left( {{zI} - \Phi_{0}} \right)}^{- 1}\quad {A\left( {{zI} - \Phi} \right)}{R(z)}} \right|}}}\quad} & (46) \end{matrix}$

where the geometric series in equation (46) $\left( {\left( \frac{\Phi_{0}}{z} \right)^{0} + \left( \frac{\Phi_{0}}{z} \right)^{1} + \left( \frac{\Phi_{0}}{z} \right)^{2} + \ldots} \right)$

converges to

z(zI−Φ₀)⁻¹.

The resulting bounds are therefore given as

|cz(zI−Φ₀)⁻¹A(zI−Φ)R(z)<2(cB+cΦ ₀(I+Φ₀)⁻¹B)  (47)

Pseudo Derivative Reference Estimators

Equations (24) is the pseudo derivative reference (PDR) Λ_(k+1) and equations (25) and (26) set forth above disclose a technique for estimating the PDR Λ_(k+1) in the control law of equation (23). This section describes various alternative embodiments for estimating the pseudo derivative reference Λ_(k+1) through the use of a least-mean-square algorithm.

In general, servo systems can be divided into three categories: systems where the reference is known, such as in robotic systems; systems where the reference is unknown but repeatable, such as the runout signal in disk storage systems; and systems where the reference is unknown and non-repeatable. For the first type of systems where the reference is known, the pseudo derivative reference Λ_(k+1) can be determined exactly from equation (22)

Λ_(k+1) =c(R_(k+1)−ΦR_(k))(cΓ)⁻¹.

For systems where the reference is unknown, the pseudo derivative reference Λ_(k+1) must be estimated.

One embodiment for estimating the Λ_(k+1) when the reference is unknown and repeatable is shown in FIG. 15A. In this example, the servo controller controls the motion of a read head in a disk storage system. The eccentricities of the disk storage medium 4 results in an unknown repeatable perturbation of the static reference signal 27 of FIG. 1. This perturbation is referred to as “runout” and is modelled as a noise signal 25 added 24 to the static reference signal 27 to generate the actual reference signal 18 for generating the position error phase state ε1 22. References of this type are harmonic in nature and are related to the angular velocity of the disk. They can be modelled as $\begin{matrix} {R_{k} = {{\sum\limits_{n = 1}^{m}{{An}_{k}{\sin ({nwk})}}} + {{Bn}_{k}{\cos ({nwk})}}}} & (48) \end{matrix}$

where m is an integer greater than zero, the coefficients An and Bn are unknown, and w is the angular velocity of the disk 4 as determined from a speed transducer 132 (FIG. 15A).

Referring to FIG. 15A, the plant 134 represents the elements of the recording device (actuator, read head, spindle motor, read channel, etc.). The input to the plant 134 is a control signal U_(k)generated by summing 136 the output of the sliding mode controller 26 (described above) and an estimated reference signal ER. The output of the plant is the measured position of the read head 6 relative to the surface of the disk 4. The detected angular velocity w is input into a sinusoid synthesizer 134 which synthesize sinusoidal signals (and harmonics) of the form in equation (48). An observer 138 estimates the non-measurable system phase states: the position phase state ˜Xp and the velocity phase state ˜Xv (or, alternatively, it may estimate the phase state errors). These phase states cannot be measured directly due to the effect of the unknown reference signal on these states. The input into the observer 138 is a position error generated by subtracting 139 the position signal X output by the plant 134 from the estimated position phase state ˜Xp output by the observer 138. The estimated phase states ˜Xp and ˜Xv are subtracted from static reference signals Rp and Rv at adders 140A and 140B to generated a position error phase state ε1 ^(k) and a velocity error phase state ε2 ^(k). Alternatively, for many classes of systems the higher order error phase states, such as the velocity error phase state, may be estimated by taking the derivative of the position error phase state.

The coefficients An and Bn of equation (48) are estimated by minimizing the position error phase state ε1 _(k) using a least-mean-square (LMS) algorithm. A sinusoid synthesizer 142 synthesizes the sinusoidal signal (and harmonics) of equation (48) which are input into an LMS adaptor 144. The LMS adaptor processes the sinusoidal signals output by the synthesizer 142 and the position error phase state ε1 ^(k) to compute the coefficients An and Bn of the estimated reference signal ER according to the following LMS algorithm ${ER}_{k} = {{\sum\limits_{n = 1}^{m}{{An}_{k}{\sin ({nwk})}}} + {{Bn}_{k}{\cos ({nwk})}}}$ An_(k + 1) = An_(k) + g  ɛ  1_(k)sin (nwk) Bn_(k + 1) = Bn_(k) + g  ɛ  1_(k)cos (nwk)

where g is a predetermined gain constant. This technique minimizes the position error, but leaves a residual modulation on Δ/(1+λ).

Adding the estimated reference ER to the control effort generated by the sliding mode controller 26 approximates the feed forward term Λ_(k+1) of equation (23). Therefore, the hardware shown in FIG. 5B for generating Λ_(k+1) is obviated.

Estimating the reference by minimizing the position error phase state ε1 ^(k) will induce a residual modulation on the sliding mode variable σ_(k) as it switches across the sliding line c. This is because the LMS algorithm reduces the error in the estimated reference at specific frequencies more than would be expected from the pseudo derivative reference term Λ_(k+1) alone. If no further adaptation is required, this residual modulation is acceptable; however, it is possible to attenuate the modulation in σ_(k) by minimizing the error in σ_(k) rather than minimizing the position error phase state ε1 _(k). This embodiment of the present invention is understood with reference to FIG. 15B.

FIG. 15B is essentially the same as FIG. 15A except that the LMS adaptor minimizes the error in σ_(k) rather than the position error phase state ε^(k). Thus, the LMS algorithm becomes ${ER}_{k} = {{\sum\limits_{n = 1}^{m}{{An}_{k}{\sin ({nwk})}}} + {{Bn}_{k}{\cos ({nwk})}}}$ ${err}_{k} = {\sigma_{k} - {{{sgn}\left( \sigma_{k} \right)}\frac{\Delta}{\left( {1 + \lambda} \right)}}}$ An_(k + 1) = An_(k) + g  err_(k)sin (nwk) Bn_(k + 1) = Bn_(k) + g  err_(k)cos (nwk)

where the variables Δ an λ determine the nominal switching boundary for σ_(k) as described above with reference to the discrete-time sliding mode controller 26. This method increases the magnitude of the residual position error within specification, but it essentially eliminates the modulation in σ_(k). In addition, it provides a better estimate of the pseudo derivative reference Λ_(k+1) than does the LMS method of minimizing the position error phase state ε1 _(k).

If the reference signal is completely unknown (non-repeatable), then the preferred method for estimating the reference is illustrated in FIG. 15C. This method enhances the estimate of the pseudo derivative reference Λ_(k+1) described in equations (25) and (26) by taking the local derivative and adding it to the estimate yielding an increasingly better estimate of Λ_(k+1)

Λ_(k+1)≈Λ_(k)

Λ_(k+1)≈Λ_(k)+(Λ_(k)−Λ_(k−1))

 Λ_(k+1)≈A_(k)Λ_(k)+B_(k)(Λ_(k)−Λ_(k−1))

where A and B are unknown coefficients. This technique may be generalized to a predictor of m derivatives similar to the LMS algorithms described above. In general, a one step predictor is sufficient to minimize the modulation in σ_(k). If, however, the reference has higher frequency components, then this technique is extensible to higher order derivatives. The LMS adaptor of FIG. 15C computes the coefficients A and B, and therefore the estimated reference signal ER, according to Λ_(k + 1) = U_(k − 1) − c  Φ(c  Γ)⁻¹ɛ_(k − 1) + σ_(k)(c  Γ)⁻¹ ${err}_{k} = {\sigma_{k} - {{{sgn}\left( \sigma_{k} \right)}\frac{\Delta}{\left( {1 + \lambda} \right)}}}$ A_(k + 1) = A_(k) + g₀err_(k)Λ_(k) B_(k + 1) = B_(k) + g₁err_(k)(Λ_(k) − Λ_(k − 1)) ER_(k) = A_(k)Λ_(k) − B_(k)(Λ_(k) − Λ_(k − 1))

where g₀ and g₁ are predetermined gain constants. The above algorithm uses the past error states ε_(k−1), the past control effort U_(k−1), and the current value of σ_(k) to generate the estimated reference signal ER. It does not require any a priori knowledge of the reference, and its convergence speed can be increased by increasing the convergence factors g₀ and g₁ (g₁ may be larger than g₀, corresponding to the slope of c).

Many changes in form and detail could be made without departing from the essential function of the present invention; the particular embodiments disclosed are not intended to be limiting. For instance, the discrete-time sliding mode controller may be implemented in hardware or software, and higher order phase states could be used in place of, or in addition to, the particular phase states disclosed. These and other modifications derived from the disclosed embodiment are within the intended scope of the present invention as properly construed from the following claims. 

We claim:
 1. An adaptive discrete-time sliding mode controller for controlling the motion of a mechanical apparatus as part of a controlled system, the adaptive discrete-time sliding mode controller comprising: (a) a phase state generator for generating a discrete-time phase state signal X1_(k), the discrete-time phase state signal X1_(k) comprising sample values occurring at a predetermined interval T; (b) a linear signal generator, responsive to the discrete-time phase state signal X1_(k), for generating a linear control signal; (c) a sliding mode signal generator, responsive to the discrete-time phase state signal X1_(k), for generating a sliding mode control signal; (d) an error signal generator for generating a feedback error signal in response to the sliding mode signal generator and a nominal signal; (e) an adapter for adapting a parameter of the controlled system in response to the feedback error signal; and (f) a combiner for combining the linear control signal and the sliding mode control signal to generate a motion control signal for controlling the motion of a mechanical apparatus.
 2. The adaptive discrete-time sliding mode controller as recited in claim 1, wherein: (a) the linear signal generator generates the linear control signal according to: Ψε_(k)  where Ψ is a constant and ε_(k) is a discrete-time phase state error signal generated from the discrete-time phase state signal X1_(k) and a reference signal R_(k); (b) the sliding mode signal generator generates the sliding mode control signal according to: μΔsgn(σ_(k))  where μ is a predetermined gain, Δ is a predetermined constant, σ_(k) is a sliding mode variable computed from the discrete-time phase state signal X1_(k), and sgn(x) returns the sign of x; and (c) the combiner generates the motion control signal according to: Ψε_(k)+μΔsgn(σ_(k)).
 3. The adaptive discrete-time sliding mode controller as recited in claim 1, wherein the sliding mode signal generator generates the sliding mode control signal according to: μΔsgn(σ_(k)) where μ is a predetermined gain, Δ is a predetermined constant, σ_(k) is a sliding mode variable computed from the discrete-time phase state signal X1_(k), and sgn(x) returns the sign of x.
 4. The adaptive discrete-time sliding mode controller as recited in claim 3, wherein the adapter adjusts the gain μ in response to the feedback error signal.
 5. The adaptive discrete-time sliding mode controller as recited in claim 3, wherein the sliding mode variable σ_(k) is computed according to: σ_(k) =cε_(k) where ε_(k) is a plurality of discrete-time phase state errors of the controlled system and c is a predetermined vector that defines a sliding line in a phase plane.
 6. The adaptive discrete-time sliding mode controller as recited in claim 5, wherein the plurality of discrete-time phase state errors ε_(k) comprises: (a) a position phase state error indicative of a position error of the mechanical apparatus relative to a position reference; and (b) a velocity phase state error indicative of a velocity error of the mechanical apparatus relative to a velocity reference.
 7. The adaptive discrete-time sliding mode controller as recited in claim 5, wherein the sliding mode variable σ_(k) changes sign at substantially every sample index k.
 8. The adaptive discrete-time sliding mode controller as recited in claim 7, wherein: (a) the sliding mode variable σ_(k) is substantially constrained to: −W<σ_(k)<+W; (b) W is the nominal signal; and (c) the error signal generator generates the feedback error signal in response to σ_(k).
 9. The adaptive discrete-time sliding mode controller as recited in claim 8, wherein the adapter adjusts the gain μ in response to the feedback error signal.
 10. The adaptive discrete-time sliding mode controller as recited in claim 9, wherein the adapter adjusts the gain μ in response to the feedback error signal in order to substantially constrain the sliding mode variable to: σ_(k+1)=−σ_(k).
 11. The adaptive discrete-time sliding mode controller as recited in claim 8, wherein the adapter adjusts the parameter of the controlled system such that the sliding mode variable σ_(k) is substantially constrained to: σ_(k)=λσ_(k)−Δsgn(σ_(k)) where λ is a predetermined constant and λ<1.
 12. The adaptive discrete-time sliding mode controller as recited in claim 11, wherein the adapter adjusts the parameter of the controlled system such that a magnitude of the sliding mode variable σ_(k) is substantially constrained to: |σ_(k)|≦Δ/(1+λ).
 13. The adaptive discrete-time sliding mode controller as recited in claim 12, wherein the adapter adjusts the gain μ in response to the feedback error signal in order to substantially constrain the sliding mode variable to: σ_(k+1)=−σ_(k).
 14. The adaptive discrete-time sliding mode controller as recited in claim 1, further comprising a reference estimator for estimating a reference signal 794 _(k), wherein: (a) the linear signal generator generates the linear control signal according to: Ψε_(k)  where Ψ is a matrix and ε_(k) is a discrete-time phase state error vector generated from a plurality of discrete-time phase state signals X_(k) and a reference signal matrix R_(k); (b) the sliding mode signal generator generates the sliding mode control signal according to: μΔsgn(σ_(k))  where μ is a predetermined gain, Δ is a predetermined constant, σ_(k) is a sliding mode variable computed from the discrete-time phase state signals X_(k), and sgn(x) returns the sign of x; and (c) the combiner generates the motion control signal according to: Ψε_(k)+Λ_(k+1)+μΔsgn(σ_(k)).
 15. An adaptive discrete-time sliding mode controller for controlling the motion of a mechanical apparatus as part of a controlled system, the adaptive discrete-time sliding mode controller comprising: (a) a phase state generator for generating a discrete-time phase state signal X1_(k), the discrete-time phase state signal X1_(k) comprising sample values occurring at a predetermined interval T; (b) a sliding mode signal generator, responsive to the discrete-time phase state signal X1_(k), for generating a sliding mode control signal for controlling the motion of a mechanical apparatus; (c) an error signal generator for generating a feedback error signal in response to the sliding mode signal generator and a nominal signal; and (d) an adapter for adapting a parameter of the controlled system in response to the feedback error signal.
 16. The adaptive discrete-time sliding mode controller as recited in claim 15, wherein the sliding mode signal generator generates the sliding mode control signal according to: μΔsgn(σ_(k)) where μ is a predetermined gain, Δ is a predetermined constant, σ_(k) is a sliding mode variable computed from the discrete-time phase state signal X1_(k), and sgn(x) returns the sign of x.
 17. The adaptive discrete-time sliding mode controller as recited in claim 16, wherein the adapter adjusts the gain μ in response to the feedback error signal.
 18. The adaptive discrete-time sliding mode controller as recited in claim 16, wherein the sliding mode variable σ_(k) is computed according to: σ_(k) =cε_(k) where ε_(k) is a plurality of discrete-time phase state errors of the controlled system and c is a predetermined vector that defines a sliding line in a phase plane.
 19. The adaptive discrete-time sliding mode controller as recited in claim 18, wherein the plurality of discrete-time phase state errors ε_(k) comprises: (a) a position phase state error indicative of a position error of the mechanical apparatus relative to a position reference; and (b) a velocity phase state error indicative of a velocity error of the mechanical apparatus relative to a velocity reference.
 20. The adaptive discrete-time sliding mode controller as recited in claim 18, wherein the sliding mode variable σ_(k) changes sign at substantially every sample index k.
 21. The adaptive discrete-time sliding mode controller as recited in claim 20, wherein: (a) the sliding mode variable σ_(k) is substantially constrained to: −W<σ_(k)<+W; (b) W is the nominal signal; and (c) the error signal generator generates the feedback error signal in response to σ_(k).
 22. The adaptive discrete-time sliding mode controller as recited in claim 21, wherein the adapter adjusts the gain μ in response to the feedback error signal.
 23. The adaptive discrete-time sliding mode controller as recited in claim 22, wherein the adapter adjusts the gain μ in response to the feedback error signal in order to substantially constrain the sliding mode variable to: σ_(k+1)=−σ_(k).
 24. The adaptive discrete-time sliding mode controller as recited in claim 20, wherein the adapter adjusts the parameter of the controlled system such that the sliding mode variable σ_(k) is substantially constrained to: σ_(k)=λσ_(k)−Δsgn(σ_(k)) where λ is a predetermined constant and λ<1.
 25. The adaptive discrete-time sliding mode controller as recited in claim 24, wherein the adapter adjusts the parameter of the controlled system such that a magnitude of the sliding mode variable σ_(k) is substantially constrained to: |σ_(k)|≦Δ/(1+λ).
 26. The adaptive discrete-time sliding mode controller as recited in claim 25, wherein the adapter adjusts the gain μ in response to the feedback error signal in order to substantially constrain the sliding mode variable to: σ_(k+1)=−σ_(k). 